Ancient Rome – there’s Jim Butcher’s Codex Alera series, Thomas Harlan’s Oath of Empire series, and many more.
Ancient China/India – Years of Salt and Rice is an interesting alternate history focusing on India and China with some fantasy elements. Grace of Kings by Ken Liu is a great fantasy take on the Romance of the Three Kingdoms. Here are some suggestions about fantasy novels set in India, although I should mention that there’s apparently a very lively fantasy scene in India if you can read Hindi or Urdu, etc.
Ancient Egypt – try some of these?